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Complexity of an SIR epidemic dynamics model with impulsive vaccination control. (English) Zbl 1065.92050
Summary: This paper considered an infected class – removed class (IR) epidemic model with impulsive vaccination, which may inherently oscillate. We studied the impulsive control and got conditions by which the epidemics elimination solution is globally asymptotically stable and conditions for the boundedness of the system. On the other hand, if the epidemic turns out to be endemic, we studied numerically the influence of impulsive vaccination on the periodic oscillations of the system without impulsions and found phenomena of chaos in this case.

MSC:
92D30 Epidemiology
34A37 Ordinary differential equations with impulses
93C15 Control/observation systems governed by ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology
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[1] Chen, L.; Chen, J., Nonlinear biological dynamic systems, (1993), Science Beijing, [in Chinese]
[2] Liu, W.; Levin, S.A.; Lwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. math. biol., 23, 187-204, (1986) · Zbl 0582.92023
[3] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. biol., 25, 359-380, (1987) · Zbl 0621.92014
[4] Wang, H., Existence of Hopf bifurcation periodic solution to SIRS epidemiological models with nonlinear incidence rates, J anhui agric univ no. 2, 29, 199-202, (2002), [in Chinese]
[5] Shulgin, B.; Stone, L.; Agur, Z., Pulse vaccination strategy in the SIR epidemic model, Bull math biol, 60, 1-26, (1998) · Zbl 0941.92026
[6] Zhen, J., Research on impulsive ecological and epidemical model (the doctor degree’s article), (2001), Xi’an Jiaotong Univ, [in Chinese]
[7] d’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. bios., 179, 1, 57-72, (2002) · Zbl 0991.92025
[8] Stone, L.; Shulgin, B.; Agur, Z., Theoretical examination of the pulse vaccination policy in the sir epidemic model, Math. comput. model, 31, 4-5, 207-215, (2000) · Zbl 1043.92527
[9] Lu, Z.; Chi, X.; Chen, L., The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Math comput model, 36, 9-10, 1039-1057, (2002) · Zbl 1023.92026
[10] Liu, X.; Chen, L., Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, solitions fractals, 16, 311-320, (2003) · Zbl 1085.34529
[11] Tang, S.; Chen, L., Chaos in functional response host cparasitoid ecosystem models, Chaos, solitons fractals, 13, 4, 875-884, (2002) · Zbl 1022.92042
[12] Liu, B.; Zhang, Y.; Chen, L., Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control, Chaos, solitons fractals, 22, 1, 123-134, (2004) · Zbl 1058.92047
[13] Zhang, S.; Chen, L., Chaos in three species food chain system with impulsive perturbations, Chaos, solitons fractals, 24, 1, 73-83, (2005) · Zbl 1066.92060
[14] Zhang, S.; Dong, L.; Chen, L., The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, solitons fractals, 23, 2, 631-643, (2005) · Zbl 1081.34041
[15] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[16] Parker, T.S.; Chua, L.O., Practical numerical algorithms for chaotic systems[M], (1989), Springer New York · Zbl 0692.58001
[17] Davies, B., Exploring chaos, theory and experiment, (1999), Perseus Books Reading, MA · Zbl 0959.37001
[18] Wolf, A.; Swift, J.B.; Swinney, L.; Vastano, A., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317, (1985) · Zbl 0585.58037
[19] May, R.M., Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974)
[20] May, R.M.; Oster, G.F., Bifurcations and dynamic complexity in simple ecological models, Am natur, 110, 573-599, (1976)
[21] Grebogi, C.; Ott, E.; Yorke, Ja., Crises, sudden changes in chaotic attractors and chaotic transients, Physica D, 7, 181-200, (1983)
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